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18
Buddy Systems
 Communications of the ACM
, 1974
"... Two algorithms are presented for implementing any of a class of buddy systems for dynamic storage allocation. Each buddy system corresponds to a set of recurrence relations which relate the block sizes provided to each other. Analyses of the internal fragmentation of the binary buddysystem, the ..."
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Two algorithms are presented for implementing any of a class of buddy systems for dynamic storage allocation. Each buddy system corresponds to a set of recurrence relations which relate the block sizes provided to each other. Analyses of the internal fragmentation of the binary buddysystem, the Fibonai buddy system, and the weighted buddy system are given. Comparative simulation results are also presented for internal, external and total fragmentation.
Special Determinants Found within Generalized Pascal Triangles
 The Fibonacci Quarterly
, 1973
"... That Pascal's triangle and two classes of generalized Pascal triangles, the multinomial coefficient arrays and the convolution arrays formed from sequences of sums of rising diagonals within the multinomial a r r a y s, share sequences of k X k unit determinants was shown in [ l]. Here, sequences of ..."
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Cited by 7 (2 self)
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That Pascal's triangle and two classes of generalized Pascal triangles, the multinomial coefficient arrays and the convolution arrays formed from sequences of sums of rising diagonals within the multinomial a r r a y s, share sequences of k X k unit determinants was shown in [ l]. Here, sequences of k X k determinants whose values are binomial coefficients in the k co column of Pascal's triangle or numbers raised to a power given by the (k 1) triangular numbers are explored. 1.
A New Angle on Pascal's Triangle
 The Fibonacci Quarterly
, 1968
"... There has always been such interest in the numbers in Pascal 1 s a r i t hmetic triangle » The sums along the horizontal rows are the powers of two, while the sums along the rising diagonals are the Fibonacci numbers * An early paper by Melvin Hochster [6] generalized the Fibonacci number property ..."
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There has always been such interest in the numbers in Pascal 1 s a r i t hmetic triangle » The sums along the horizontal rows are the powers of two, while the sums along the rising diagonals are the Fibonacci numbers * An early paper by Melvin Hochster [6] generalized the Fibonacci number property by
Convolution Triangles for Generalized Fibonacci Numbers." The Fibonacci Quarterly 8
"... The sequence of integers Fj = 1, F2 = 1, and F 2 = F t + F are called the Fibonacci numbers. The numbers Fj and F2 are called the starting pair and F 2 = F n + i + F i s c a e c t n e n ^ ^ ..."
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Cited by 5 (1 self)
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The sequence of integers Fj = 1, F2 = 1, and F 2 = F t + F are called the Fibonacci numbers. The numbers Fj and F2 are called the starting pair and F 2 = F n + i + F i s c a e c t n e n ^ ^
Some Generating Functions
 Duke Math. J
, 1963
"... With an arbitrary sequence of (complex) numbers (a i = {a0,ai,a2, ' • y we associate the (formal) power series (1.1) a(x) = £ a x 11. n=0 ..."
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With an arbitrary sequence of (complex) numbers (a i = {a0,ai,a2, ' • y we associate the (formal) power series (1.1) a(x) = £ a x 11. n=0
DIAGONAL SUMS OF THE TRINOMIAL TRIANGLE
"... In an earlier paper [ l] , a method was given for finding the sum of terms along any rising diagonals in any polynomial coefficient array, given by (1 + x + x 2 + • • + x 1 " " 1) 11, n = 0, 1, 2, • • • , r> 2, which sums generalized the numbers u(n; p,q) of Harris and Styles [2] , [ 3]. In thi ..."
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Cited by 1 (1 self)
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In an earlier paper [ l] , a method was given for finding the sum of terms along any rising diagonals in any polynomial coefficient array, given by (1 + x + x 2 + • • + x 1 " " 1) 11, n = 0, 1, 2, • • • , r> 2, which sums generalized the numbers u(n; p,q) of Harris and Styles [2] , [ 3]. In this paper, an explicit solution of the general case for the trinomial triangle is derived. If we write only the coefficients appearing in the expansions of the trinomial (1 + x + x 2) , we have 1
LINEARLY RECURSIVE SEQUENCES OF INTEGERS
"... As harmless as it may appear, the Fibonacci sequence has provoked a remarkable amount of research. It seems that there is no end to the results that may be derived from the basic definition F ^ 0 = F _,_., + F and F0 = 0 and F4 = 1, ..."
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As harmless as it may appear, the Fibonacci sequence has provoked a remarkable amount of research. It seems that there is no end to the results that may be derived from the basic definition F ^ 0 = F _,_., + F and F0 = 0 and F4 = 1,
ON A PARTITION OF GENERALIZED FIBONACCI NUMBERS
"... As a continuation of results in [ 4] , this paper deals with the concept of minimal and maximal representations of positive integers as sums of generalized Fibonacci numbers (G, F. N.) defined below and presents a partition of the G. F. N. in relation to either minimal or maximal representation. Con ..."
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As a continuation of results in [ 4] , this paper deals with the concept of minimal and maximal representations of positive integers as sums of generalized Fibonacci numbers (G, F. N.) defined below and presents a partition of the G. F. N. in relation to either minimal or maximal representation. Consider the sequence {F}, where (1) and Fi = F2 = •• • = F r = 1, r A 2 F, = F, + F, , t A r t tl tr Obviously, the sequence gives rise to the sequence of Fibonacci numbers for r = 2. For this reason, we call {F,} a sequence of G.F.N. Clearly {F,} is a special case of Daykin's Fibonacci sequence [3], as well as of Harris and Styles f sequence [6]. We remark that it is possible to express any positive integer N as a sum of distinct F. T s, subject to the condition that F1} F2, • • ° , F are not used in any sum (reference: Daykin's paper [3]). In other words, we can have (2) N = £ a i F i i = r with a = 1 and a. • = 1 or 0, r 4 i A s. Here s is the largest integer such that F is involved in the sum. s Definition 1: In case (2) is satisfied, the vector (a, a,.,•••., a) of — r r+i s elements 1 or 0 with a = 1, is called a representation of N in {F}, s z having its index as s. Definition 2: A representation of N in {F,} is said to be minimal or maximal according to as a.a.,. = 0 or a. + a.,. ^ 1, for all i = ^ r and i = l l+j 1 1 + 3 l, 2, * ' , r l. 22 1968 ON A PARTITION OF GENERALIZED FIBONACCI NUMBERS 23 Definition 2 is just an extension of that by Ferns [ 4] to r> 2. Now, we state some results in the forms of lemmas, to be used subsequently, Lemma 1: (i) Every positive integer N has a unique minimal representation; (ii) The index of the minimal representation of N (F < N < F, n> r) is n, If N = F, then a = a, = « = a = 0. n r r+i ni Lemma 2: (i) Every positive integer N has a unique maximal representation: (ii) The index of the maximal representation o f N ( F + r < N < F + r9 n> r) is n. If N = F, r, then a = a, = • • • = a = 1» / n + r r r + 1 n_j Lemma 3. If the minimal representation of is N< = F n + r rN <Fu < W < F u + 1, n> r, r <: u < n 1) (a, a.. • • •, a), \ r » r+1 » u then the maximal representation of N ( F ^ F ^ ~ r < N < F J _ F r) n+r u+i n+r u is and conversely.
1 1 1 1 1
"... There are many ways in which one can select a square array from Pascal's triangle which will have a determinant of value one. What is surprising is that two classes of generalized Pascal triangles which arose in [l] also have this property: the multinomial coefficient triangles and the convolution t ..."
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There are many ways in which one can select a square array from Pascal's triangle which will have a determinant of value one. What is surprising is that two classes of generalized Pascal triangles which arose in [l] also have this property: the multinomial coefficient triangles and the convolution triangles formed from sequences which.are found as the sums of elements appearing on rising diagonals within the binomial and multinomial coefficient triangles. The generalized Pascal triangles also share sequences of k x k determinants whose values are successive binomial coefficients in the k column of PascaPs triangle. 1. UNIT DETERMINANTS WITHIN PASCAL'S TRIANGLE When Pascal's triangle is imbedded in matrices throughout this paper, we will number the rows and columns in the usual matrix notation, with the leftmost column the first column. If we refer to Pascal's triangle itself9 then the leftmost column is the zero column, and the top row is the zero row. First we write Pascal's triangle in rectangular form as the n x n matrix P = (p.,). (1.1) 1
#•*•«•aa anX. DIAGONAL SUMS IN THE HARMONIC TRIANGLE
"... The great geographical distance between us prevented us from seeing one another very often. I did, on my way to lecturing in Hawaii, stop off to see Vern, and I spent a few days with him a couple of years later when I lectured along the California coast. He once visited me at the University of Maine ..."
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The great geographical distance between us prevented us from seeing one another very often. I did, on my way to lecturing in Hawaii, stop off to see Vern, and I spent a few days with him a couple of years later when I lectured along the California coast. He once visited me at the University of Maine, when, representing his university, he came as a delegate to a national meeting of Phi Kappa Phi (an academic honorary that was founded at the University of Maine). For almost four decades I had the enormous pleasure of Vern T s friendship, and bore the flattering title, generously bestowed upon me by him, of his "mathematical mentor." In mathematics, Vern was a skylark, and I regret, far more than I can possibly express, the sad fact that we now no longer will hear further songs by him. But, oh, on the other hand, how privileged I have been; I heard the skylark when he first started to sing. HaJJL to tkdd, bLutkd SpvUAl Btnd tkou n&ve/i wejvt,